Optimal. Leaf size=113 \[ -\frac{2 c^2 (b B-A c)}{b^4 \sqrt{x}}-\frac{2 c^{5/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{9/2}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{2 (b B-A c)}{5 b^2 x^{5/2}}-\frac{2 A}{7 b x^{7/2}} \]
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Rubi [A] time = 0.0676766, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {781, 78, 51, 63, 205} \[ -\frac{2 c^2 (b B-A c)}{b^4 \sqrt{x}}-\frac{2 c^{5/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{9/2}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{2 (b B-A c)}{5 b^2 x^{5/2}}-\frac{2 A}{7 b x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 781
Rule 78
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x}{x^{7/2} \left (b x+c x^2\right )} \, dx &=\int \frac{A+B x}{x^{9/2} (b+c x)} \, dx\\ &=-\frac{2 A}{7 b x^{7/2}}+\frac{\left (2 \left (\frac{7 b B}{2}-\frac{7 A c}{2}\right )\right ) \int \frac{1}{x^{7/2} (b+c x)} \, dx}{7 b}\\ &=-\frac{2 A}{7 b x^{7/2}}-\frac{2 (b B-A c)}{5 b^2 x^{5/2}}-\frac{(c (b B-A c)) \int \frac{1}{x^{5/2} (b+c x)} \, dx}{b^2}\\ &=-\frac{2 A}{7 b x^{7/2}}-\frac{2 (b B-A c)}{5 b^2 x^{5/2}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}+\frac{\left (c^2 (b B-A c)\right ) \int \frac{1}{x^{3/2} (b+c x)} \, dx}{b^3}\\ &=-\frac{2 A}{7 b x^{7/2}}-\frac{2 (b B-A c)}{5 b^2 x^{5/2}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{2 c^2 (b B-A c)}{b^4 \sqrt{x}}-\frac{\left (c^3 (b B-A c)\right ) \int \frac{1}{\sqrt{x} (b+c x)} \, dx}{b^4}\\ &=-\frac{2 A}{7 b x^{7/2}}-\frac{2 (b B-A c)}{5 b^2 x^{5/2}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{2 c^2 (b B-A c)}{b^4 \sqrt{x}}-\frac{\left (2 c^3 (b B-A c)\right ) \operatorname{Subst}\left (\int \frac{1}{b+c x^2} \, dx,x,\sqrt{x}\right )}{b^4}\\ &=-\frac{2 A}{7 b x^{7/2}}-\frac{2 (b B-A c)}{5 b^2 x^{5/2}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{2 c^2 (b B-A c)}{b^4 \sqrt{x}}-\frac{2 c^{5/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0140488, size = 44, normalized size = 0.39 \[ \frac{2 \left (\, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};-\frac{c x}{b}\right ) (7 A c x-7 b B x)-5 A b\right )}{35 b^2 x^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 126, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{7\,b}{x}^{-{\frac{7}{2}}}}+{\frac{2\,Ac}{5\,{b}^{2}}{x}^{-{\frac{5}{2}}}}-{\frac{2\,B}{5\,b}{x}^{-{\frac{5}{2}}}}-{\frac{2\,A{c}^{2}}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}+{\frac{2\,Bc}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}+2\,{\frac{{c}^{3}A}{{b}^{4}\sqrt{x}}}-2\,{\frac{{c}^{2}B}{{b}^{3}\sqrt{x}}}+2\,{\frac{A{c}^{4}}{{b}^{4}\sqrt{bc}}\arctan \left ({\frac{\sqrt{x}c}{\sqrt{bc}}} \right ) }-2\,{\frac{{c}^{3}B}{{b}^{3}\sqrt{bc}}\arctan \left ({\frac{\sqrt{x}c}{\sqrt{bc}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61672, size = 548, normalized size = 4.85 \begin{align*} \left [-\frac{105 \,{\left (B b c^{2} - A c^{3}\right )} x^{4} \sqrt{-\frac{c}{b}} \log \left (\frac{c x + 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 2 \,{\left (15 \, A b^{3} + 105 \,{\left (B b c^{2} - A c^{3}\right )} x^{3} - 35 \,{\left (B b^{2} c - A b c^{2}\right )} x^{2} + 21 \,{\left (B b^{3} - A b^{2} c\right )} x\right )} \sqrt{x}}{105 \, b^{4} x^{4}}, \frac{2 \,{\left (105 \,{\left (B b c^{2} - A c^{3}\right )} x^{4} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) -{\left (15 \, A b^{3} + 105 \,{\left (B b c^{2} - A c^{3}\right )} x^{3} - 35 \,{\left (B b^{2} c - A b c^{2}\right )} x^{2} + 21 \,{\left (B b^{3} - A b^{2} c\right )} x\right )} \sqrt{x}\right )}}{105 \, b^{4} x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 73.191, size = 326, normalized size = 2.88 \begin{align*} \begin{cases} \tilde{\infty } \left (- \frac{2 A}{9 x^{\frac{9}{2}}} - \frac{2 B}{7 x^{\frac{7}{2}}}\right ) & \text{for}\: b = 0 \wedge c = 0 \\\frac{- \frac{2 A}{9 x^{\frac{9}{2}}} - \frac{2 B}{7 x^{\frac{7}{2}}}}{c} & \text{for}\: b = 0 \\\frac{- \frac{2 A}{7 x^{\frac{7}{2}}} - \frac{2 B}{5 x^{\frac{5}{2}}}}{b} & \text{for}\: c = 0 \\- \frac{2 A}{7 b x^{\frac{7}{2}}} + \frac{2 A c}{5 b^{2} x^{\frac{5}{2}}} - \frac{2 A c^{2}}{3 b^{3} x^{\frac{3}{2}}} + \frac{2 A c^{3}}{b^{4} \sqrt{x}} - \frac{i A c^{3} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{c}} + \sqrt{x} \right )}}{b^{\frac{9}{2}} \sqrt{\frac{1}{c}}} + \frac{i A c^{3} \log{\left (i \sqrt{b} \sqrt{\frac{1}{c}} + \sqrt{x} \right )}}{b^{\frac{9}{2}} \sqrt{\frac{1}{c}}} - \frac{2 B}{5 b x^{\frac{5}{2}}} + \frac{2 B c}{3 b^{2} x^{\frac{3}{2}}} - \frac{2 B c^{2}}{b^{3} \sqrt{x}} + \frac{i B c^{2} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{c}} + \sqrt{x} \right )}}{b^{\frac{7}{2}} \sqrt{\frac{1}{c}}} - \frac{i B c^{2} \log{\left (i \sqrt{b} \sqrt{\frac{1}{c}} + \sqrt{x} \right )}}{b^{\frac{7}{2}} \sqrt{\frac{1}{c}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16367, size = 140, normalized size = 1.24 \begin{align*} -\frac{2 \,{\left (B b c^{3} - A c^{4}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} b^{4}} - \frac{2 \,{\left (105 \, B b c^{2} x^{3} - 105 \, A c^{3} x^{3} - 35 \, B b^{2} c x^{2} + 35 \, A b c^{2} x^{2} + 21 \, B b^{3} x - 21 \, A b^{2} c x + 15 \, A b^{3}\right )}}{105 \, b^{4} x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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